6924
/www/nrich/html/content/id/6924/
1
2011-02-01T00:00:01
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Nearly all of us have made table patterns on hundred squares, that
is $10$ by $10$ grids. Some tables made vertical lines, some made
diagonal lines and some patterns ranging across the whole space.
Hundred squares are $10$ by $10$ grids. In this problem we will
call these "$10$ grids".<br></br>
<div style="text-align: center;"><mdo:image width="478" height="144" alt="table patterns on 10s" src="tens.png"></mdo:image></div>
<br></br>
What numbers made which sort of patterns and why?<br></br>
<br></br>
This problem looks at the patterns on differently sized square
grids. These are from $4$ grids (that is a $4$ by $4$ grid) to $9$
grids.<br></br>
<br></br>
These are patterns on a $7$, a $5$, an $8$ and on a $6$ grid:<br></br>
<div style="text-align: center;"><mdo:image width="415" height="115" alt="different grids" src="odds.png"></mdo:image></div>
<br></br>
What tables made these patterns? Can you think why they made them
like that?<br></br>
<br></br>
Perhaps this is the time to experiment for yourself. You can use
grids drawn on squared paper or use <a href="/content/id/6924/4-9Grids.pdf">this sheet</a>.<br></br>
Can you discover what makes vertical and diagonal lines on the
different grids and what makes the various patterns. Can you make
the checked pattern? What table do you need to use on what kind of
grid?<br></br>
<br></br>
Here are the top parts of some grids.<br></br>
Can you identify what table on what grid have been used to make
them?<br></br>
<br></br>
<div style="text-align: center;"><mdo:image width="466" height="67" alt="tops of different grids" src="gridTops.png"></mdo:image></div>
<br></br>
Here are some parts of various grids. This time we have not shown
the edges of the grids. Can you identify what tables on what grids
could have been used to make these patterns?<br></br>
<br></br>
<div style="text-align: center;"><mdo:image width="503" height="72" alt="different grids" src="midGrids.png"></mdo:image></div>
<br></br>
There may be more than one answer.<br></br></mdoxml>
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<p class="editorial">Rajeev
from Fair Field Junior looked at the first row of grids in the
problem and said:</p>
<div>In any grid where every
$4$th square is shaded then it will contain multiples of $4$ and so
on. In the $7$ grid as every $3$rd square is shaded it has
multiples of $3$, in the $5$ grid where every $2$nd square is
shaded you get multiples of $2$ and in the $8$ grid where every
$5$th square is shaded you get multiples of $5$ and if you get a
negative diagonal line in the $6$ grid starting from the $1$st
square in the $2$nd row you get multiples of $7$.</div>
<br></br>
<p class="editorial">For the
row of grids where you could only see the top of them, Rajeev
said:</p>
<div>If it's a $7$ grid and
every $4$th square is shaded it contains multiples of $4$. I have
also noticed that the lines run parallel diagonally and the
difference between the shaded squares is $3$ in the $7$ grid so
every diagonal line will contain multiples of $4$.</div>
<div>If the lines run
diagonally parallel and the difference between shaded squares in a
row is $2$ then all the diagonal lines will have multiples of
$3$.</div>
<div>Where every $2$nd square
is shaded then it contains multiples of $2$ and as explained the
last column in any grid will contain multiples of that grid.</div>
<br></br>
<p class="editorial">Rajeev
goes on to say that where the edges are not shown you can still
identify the tables:</p>
<div>In the $1$st grid the
line is negative running diagonally and if more than $4$ squares
are shaded it must be a $6$ grid or more and the tables could be
$7$ or more.</div>
<br></br>
<p class="editorial">What can
we say about the relationship between the grid size and times table
in this first grid, I wonder?</p>
<div>In the $2$nd grid as the
line is positive and running diagonally and at least $4$ squares
are shaded the grid must be at least $5$ by $5$ or more.</div>
<br></br>
<div>With the $3$rd pattern it
is a $9$ grid and contains $6$ times table, because the difference
between the squares horizontally is $5$ and vertically $1$. The
grid must be a multiple of $3$.</div>
<br></br>
<div>With the $4$th pattern it
is an $8$ grid and has multiples of $3$. The grid could also be
$11$, $14$, $17$, $20$, $23$ ... .</div>
<br></br>
<div>The last grid must be $8$
or more.</div>
<br></br>
<p><span class="editorial">I
wonder what else we can say about the last grid? Fantastic work,
Rajeev. You've explained your thinking very clearly.</span></p>
<p class="editorial">Grace,
Libby, Chloe-Anne and Becky from Maldon Primary School looked at
the patterns of tables on differently-sized grids. Chloe-Anne
pointed out:</p>
$2$x on an even grid will go down in columns and on an odd grid it
is like a checkerboard.<br></br>
<br></br>
<p class="editorial">Becky
noticed:</p>
If you have a number and it's that type of number grid eg $7$ by
$7$ grid means that you would get the $7$ times table going
downwards or eg $5$ by $5$ grid you would get the $5$ times table
going downwards in a straight line.<br></br>
<br></br>
<p class="editorial">So, I
think Becky is saying that when you create the pattern of the times
table that is the same as the size of the grid, you get a straight
line going downwards, or vertically. Well spotted!</p>
<br></br>
<p class="editorial">Mrs
Cresswell's Maths Group from Manor School, Didcot wrote:</p>
We really enjoyed trying the times tables on $10$x$10$ grids and we
found a pattern we could describe for all the times tables except
the $7$ times table!<br></br>
We noticed that the $9$x table had a diagonal pattern because the
grid was one larger than $9$, so the number we coloured in ended up
one further back on the row below each time.<br></br>
We tried this out with other numbers that were one less than the
size of the grid, and found that this always makes a diagonal
pattern. We drew a picture of the $4$x table on a $5$x$5$ grid and
the $6$x table on a $7$x$7$ grid to show you:<br></br>
<br></br>
<table border="1">
<tbody>
<tr>
<td><mdo:image width="205" height="198" src="4sOn5grid.gif" alt=""></mdo:image></td>
<td><mdo:image width="265" height="265" src="6sOn7grid.gif" alt=""></mdo:image></td>
</tr>
</tbody>
</table>
<br></br>
<br></br>
We wanted to make a diagonal going the other way, so we tried times
tables that were one bigger than the grid. We did the $11$x table
on a $10$x$10$ grid, and got a diagonal going down from left to
right, because the tens and the units got bigger each time.<br></br>
<br></br>
<mdo:image width="325" height="325" src="9s11s.gif" alt=""></mdo:image><br></br>
<br></br>
<p class="editorial">Thank you
to everyone who submitted solutions to this problem. There's so
much to explore here, isn't there?</p>
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<h2>Table Patterns Go Wild!</h2>
<br></br>
Nearly all of us have made table patterns on hundred squares, that is $10$ by $10$ grids. Some tables made vertical lines, some made diagonal lines and some patterns ranging across the whole space. Hundred squares are $10$ by $10$ grids. In this problem we will call these "$10$ grids".<br></br>
<div style="text-align: center;"><mdo:image alt="table patterns on 10s" height="144" src="tens.png" width="478"></mdo:image></div>
<br></br>
What numbers made which sort of patterns and why?<br></br>
<br></br>
This problem looks at the patterns on differently sized square grids. These are from $4$ grids (that is a $4$ by $4$ grid) to $9$ grids.<br></br>
<br></br>
These are patterns on a $7$, a $5$, an $8$ and on a $6$ grid:<br></br>
<div style="text-align: center;"><mdo:image alt="different grids" height="115" src="odds.png" width="415"></mdo:image></div>
<br></br>
What tables made these patterns? Can you think why they made them like that?<br></br>
<br></br>
Perhaps this is the time to experiment for yourself. You can use grids drawn on squared paper or use <a href="/content/id/6924/4-9Grids.pdf">this sheet</a>.<br></br>
Can you discover what makes vertical and diagonal lines on the different grids and what makes the various patterns. Can you make the checked pattern? What table do you need to use on what kind of grid?<br></br>
<br></br>
Here are the top parts of some grids.<br></br>
Can you identify what table on what grid have been used to make them?<br></br>
<br></br>
<div style="text-align: center;"><mdo:image alt="tops of different grids" height="67" src="gridTops.png" width="466"></mdo:image></div>
<br></br>
Here are some parts of various grids. This time we have not shown the edges of the grids. Can you identify what tables on what grids could have been used to make these patterns?<br></br>
<br></br>
<div style="text-align: center;"><mdo:image alt="different grids" height="72" src="midGrids.png" width="503"></mdo:image></div>
<br></br>
There may be more than one answer.</div>
<br></br>
<h3><span style="font-weight: bold;">Why do this problem?</span></h3>
<div><a href="http://nrich.maths.org/6924&part=">This problem</a> helps to extend learners' familiarity with factors as well as providing interesting work on multiplication tables. They can make hypotheses and test them in a simple and easily understood environment. It provides an opportunity to introduce the idea of a letter standing for "any number".</div>
<br></br>
<div>It can also add to their ideas of pattern and design.</div>
<br></br>
<h3>Possible approach</h3>
<div>If learners have not done many table patterns on hundred squares you could start by doing some of these. It is important that they do not see these as a series of straight lines, so it is advisable to get them to do the three times and four times tables before they embark on the two and five times. They should be asked why the tables of twos and fives make straight lines. Of course the word
"factor" might pop up here and there.</div>
<br></br>
<div>If plenty of work has already been done on in this area ask what makes straight vertical lines and why. Next give out <span style="font-weight: bold;">unnumbered</span> $10$ grids (on <a href="/content/id/6924/10Grids.pdf">this sheet</a> or on squared paper) and get different learners to do different numbers so all are done, asking them to predict what sort of pattern they think they will
get. It is advisable to suggest putting crosses on the appropriate squares as colouring them in can take a long time!</div>
<br></br>
<div>After this introduce the different grids. They can be found on <a href="/content/id/6924/4-9Grids.pdf">this sheet</a>. Learners could work in pairs to share each other's findings and also discuss their ideas with a partner.</div>
<br></br>
<div>There are two photocopiable "problem sheets" for learners to identify the tables and grids. These have more on them than the ones given in the problem. <a href="/content/id/6924/6924A.pdf">This one</a> (A) has clearly defined grids and so is much easier than <a href="/content/id/6924/6924B.pdf">this one</a> (B) which is a real challenge. In sheet B the sizes of the grids are more ambiguous
and which may, therefore, have more than one answer. Squared paper is useful in investigating these.</div>
<br></br>
<div>At the end of the lesson learners should be asked to say what makes the different patterns. The factors of the grid number should be discussed and also the idea of one more than and one less than. It is possible here to introduce or practise using a letter to stand for "any number".</div>
<br></br>
<div>You could also discuss what makes an interesting all-over pattern.</div>
<br></br>
<h3>Key questions</h3>
<div>What kind of pattern do you think you are going to get?</div>
<div>Which tables will make vertical lines?</div>
<div>Why do you think this is?</div>
<div>Which tables will make diagonal lines?</div>
<div>Why do you think this is?</div>
<div>What kind of grid do you need to make a pattern of checks?</div>
<div>What do you think makes an interesting all-over pattern?</div>
<br></br>
<h3>Possible extension</h3>
<div>If they finish early learners could make some similar "puzzles" on different grids for others to do.</div>
<br></br>
<div>Those who are really confident with the grids up to ten could predict what they will get and then explore eleven and twelve grids (which can be drawn on squared paper). They could also be expected to express their findings in algebraic terms where possible.</div>
<br></br>
<h3>Possible support</h3>
Suggest concentrating on the patterns on hundred squares and $10$ grids.<br></br>
<br></br>
<br></br>
<br></br>
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<mdoxml version="1.0">Try the various patterns out on squared paper.<br></br></mdoxml>
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Table patterns go wild!
Nearly all of us have made table patterns on hundred squares, that
is 10 by 10 grids. This problem looks at the patterns on
differently sized square grids.
Using, Applying and Reasoning about Mathematics
Selecting and using information
Using, Applying and Reasoning about Mathematics
Making and testing hypotheses
Numbers and the Number System
Properties of numbers
Numbers and the Number System
Factors and multiples
Algebra
Introducing algebra
Admin
Upper primary mapping document